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On Dwork’s $p$-adic formal congruences theorem and hypergeometric mirror maps


About this Title

E. Delaygue, T. Rivoal and J. Roques

Publication: Memoirs of the American Mathematical Society
Publication Year: 2017; Volume 246, Number 1163
ISBNs: 978-1-4704-2300-1 (print); 978-1-4704-3635-3 (online)
DOI: https://doi.org/10.1090/memo/1163
Published electronically: October 13, 2016
Keywords:Dwork’s theory, generalized hypergeometric functions, $p$-adic analysis, integrality of mirror maps

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Statements of the main results
  • Chapter 3. Structure of the paper
  • Chapter 4. Comments on the main results, comparison with previous results and open questions
  • Chapter 5. The $p$-adic valuation of Pochhammer symbols
  • Chapter 6. Proof of Theorem 4
  • Chapter 7. Formal congruences
  • Chapter 8. Proof of Theorem 6
  • Chapter 9. Proof of Theorem 9
  • Chapter 10. Proof of Theorem 12
  • Chapter 11. Proof of Theorem 8
  • Chapter 12. Proof of Theorem 10
  • Chapter 13. Proof of Corollary 14

Abstract


Using Dwork's theory, we prove a broad generalization of his famous -adic formal congruences theorem. This enables us to prove certain -adic congruences for the generalized hypergeometric series with rational parameters; in particular, they hold for any prime number and not only for almost all primes. Furthermore, using Christol's functions, we provide an explicit formula for the âĂIJEisenstein constantâĂİ of any hypergeometric series with rational parameters.As an application of these results, we obtain an arithmetic statement âĂIJon averageâĂİ of a new type concerning the integrality of Taylor coefficients of the associated mirror maps. It contains all the similar univariate integrality results in the literature, with the exception of certain refinements that hold only in very particular cases.

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